Tag: angles

Variation theory of learning was developed by Ference Marton of the University of Gothenburg. One of its basic tenets is that learning is always directed at something – the object of learning (phenomenon, object, skills, or certain aspects of reality) and that learning must result in a qualitative change in the way of seeing this “something” (Ling & Marton, 2011). Variation theory sees learning as the ability to discern different features or aspects of what is being learned. It postulates that the conception one forms about something or how something is understood is related to the aspects of the object one notices and focuses on.

Here’s an example: In linear equations you want your students to learn that a linear equation in one unknown can only have one root while an equation with two unknowns can have infinitely many roots.  You also want them to learn that in an equation of one unknown, the root is represented by x only while in equation with two unknowns, the root is represented by an ordered pair of x and y. It is also important that students will see that while both roots can be represented by a point, the root of the equation in one unknown can be plotted in a number line or one-dimensional axis while the root of the equation in two unknowns are plotted in two-dimensional coordinate axes. Will the students discern these particular differences between the roots of the two types of equation in the natural course of teaching linear equations or should you so design the lesson so that students will focus on these differences? Variation theory tells you, yes, you should.

At the World Association of Lesson Studies (WALS) conference in HongKong in 2010 most of the lesson studies presented were informed by variation theory. The teachers reported that students achievement showed significant increases in the post test. Everybody seemed to be happy about it. I think it is not only because of its effect on achievement but it also gave the teachers a framework for structuring their lesson particularly on the design and sequencing of tasks. This sounds very simple but it is actually challenging. The challenge is in identifying the critical feature for a particular object of learning – what is it they need to vary and what needs to remain invariant in the students experiences. Variation theory asserts that change in conception can occur by highlighting critical elements of the object of learning and creating variation in these while all other elements are held constant.

Variation theory directs the teacher to focus on the critical aspect of the object of learning (a math concept, for example), identify differing level of conceptions, and from each of these conceptions identify the critical elements (core ideas) which needed to be varied and those that will remain invariant. In mathematics, these invariants are usually the properties of the concept. In the case of the angles for example, in order for students to have a ‘full’ understanding of this concept they needed to experience it in different forms – the two-line angles, the one-line angles, and the no-line angles.

What they need to learn (abstract) from these is that they all consist of two linear parts (even if they are not visible) and they cross or meet at a point and that the relative inclination of the two parts has some significance – it defines the sharpness of the corner or the their openness. Given these, the teacher now has to design the lesson/ tasks that will provide the necessary variation of learning experiences. You can read my post Angles aren’t that Easy to See for further explanation about understanding angles. Check also my post on how to select and sequence examples to see how variation theory is useful for thinking about examples.

Teachers must always remember however that “even if they aware of the need for the appropriate pattern of variation and invariance, quite a bit of ingenuity may be required to bring it about. Providing the necessary conditions for learning does not guarantee that learning will take place. It is the students’ experience of the conditions that matters. Some students will learn even though the necessary conditions are not provided in class. This may be because such conditions were available in the students’ past, and some students are able to recall these experiences to provide a contrast with what they experience in class. But, as teachers we should not leave learning to happen by chance, and we should strive to provide the necessary conditions to the extent that we are able” (Ling & Marton, 2011). I think we should also remember that the way the learners are engaged is a big factor in learning. You may have addressed the critical feature through examples with appropriate pattern of variation but if this was done by telling, learning may still be limited and superficial.

Another useful guide for effecting learning is creating cognitive conflict. Click Using cognitive conflict to teach solving inequalities to see a sample lesson.